Computation of the Lennard Jones curve

In this exercise you will compute the Lennard-Jones energy curve for a system of two Ar atoms.
In Part I you find the instructions for computing the energy of two Ar atoms at a distance $r=3.00 Å$.
In Part II you find the instructions for getting the energy profile as a function of $r$.
Additonal parameters for Xe and combination rules to obtain new parameters are provided in Part III and IV.

Part I: Single Point (Energy) calculation

In this section a commented CP2K input example for a single point calculation is provided.
Comments are added and signaled with '!'.

If you get the closing Banner you know that cp2k worked. The following line tells you the result:

ENERGY| Total FORCE_EVAL ( FIST ) energy (a.u.): 0.003617048870059

This is the energy (in Hartree) for a system of 2 Ar atoms at distance $ r=3.00 Å$

Note, that in the input-file EPSILON is given in units of Kelvin, whereas in the output the energy is printed in Hartree, which is the unit of energy in the system of atomic units (a.u.). To convert from Kelvin to Hartree you have to multiply with the Boltzmann constant $ k_\text{b} = 3.1668154 \cdot 10^{-6} \frac{E_\text{H}}{\text{K}} $ .

Part II: Computation of the LJ energy curve

In this section a few scripts to get the LJ energy profiles are presented.

1. Step

In order to get a good profile, a set of energy values as a function of the interatomic distance is needed. You can use the energy.inp input file and change the Ar coordinates in order to get different starting distances.

The output file will be rewritten every time you run a calculation, unless you change its name.

To do so:

$ mv energy.out energy_dist3A.out

If you run multiple calculations, it is always good to keep track of what you have done by producing an input and an output for every distance you are planning to run.

For doing so:

$ cp energy.inp energy_dist2A.inp

then edit the input file with the new coordinates (e.g. 2 Å).
you can now run CP2K and produce the output file:

$ cp2k.popt -i energy_dist2A.inp -o energy_dist2A.out

2. Step

When you have tested a few distances, you can produce a table looking like:

Input file

Distance (Å)

Energy (Eh)

energy_dist1A.inp

1

…

energy_dist2A.inp

2

…

energy_dist3A.inp

3

…

…

…

…

This is the Lennard Jones energy curve for two Ar atoms.
By using any plotting program you can now get a representation of the energy profile.

3. Step

Here are reported the LJ parameters for Xe atoms. Those are to replace the Ar parameters in the input file, along with your Xe coordinates that have to replace the Ar coordinates. A new LJ curve for Xe atoms can be now generated.

4. Step

Here are reported the combination rules for pairs unlike pairs, i.e. for pairs of non identical atoms.
Once generated the ε and σ parameters for the couple Ar/Xe, generate once more the LJ dissociation curve.
Compare the “mixed” curve to the two “pure” curves and report the position and depth of the minimum.